# Hilbert space and ground state structure of bilayer quantum Hall systems at $\nu=2/\lambda$

## TL;DR

This paper investigates the ground state structure and phase diagram of bilayer quantum Hall systems at fractional filling factors, using both semiclassical and quantum numerical methods, revealing phase transitions and energy level crossings dependent on system parameters.

## Contribution

It extends the analysis of bilayer quantum Hall systems to fractional fillings with arbitrary flux quanta, combining mean-field and numerical methods to explore phase transitions and energy spectra.

## Key findings

- Identification of three quantum phases: spin, canted, and ppin.
- Detection of $ ext{lambda}$ energy level crossings in the canted phase.
- Energy band structures depend on tunneling and quantum numbers.

## Abstract

We analyze the Hilbert space and ground state structure of bilayer quantum Hall (BLQH) systems at fractional filling factors $\nu=2/\lambda$ ($\lambda$ odd) and we also study the large $SU(4)$ isospin-$\lambda$ limit. The model Hamiltonian is an adaptation of the $\nu=2$ case [Z.F. Ezawa {\it et al.}, Phys. Rev. {B71} (2005) 125318] to the many-body situation (arbitrary $\lambda$ flux quanta per electron). The semiclassical regime and quantum phase diagram (in terms of layer distance, Zeeeman, tunneling, etc, control parameters) is obtained by using previously introduced Grassmannian $\mathbb{G}^4_{2}=U(4)/[U(2)\times U(2)]$ coherent states as variational states. The existence of three quantum phases (spin, canted and ppin) is common to any $\lambda$, but the phase transition points depend on $\lambda$, and the instance $\lambda=1$ is recovered as a particular case. We also analyze the quantum case through a numerical diagonalization of the Hamiltonian and compare with the mean-field results, which give a good approximation in the spin and ppin phases but not in the canted phase, where we detect exactly $\lambda$ energy level crossings between the ground and first excited state for given values of the tunneling gap. An energy band structure at low and high interlayer tunneling (spin and ppin phases, respectively) also appears depending on angular momentum and layer population imbalance quantum numbers.